Eigenvalues inequalities for convex and logconvex functions

TL;DR

This paper explores inequalities involving eigenvalues of Hermitian matrices with convex and log-convex functions, providing simplified proofs of key inequalities like Bhatia-Kittaneh and Naimark dilation theorem.

Contribution

It offers new, simplified proofs of important eigenvalue inequalities for convex functions and Hermitian matrices, enhancing understanding and application.

Findings

Simplified proof of Bhatia-Kittaneh inequality

Simplified proof of Naimark dilation theorem

Extension of eigenvalue inequalities to (log) convex functions

Abstract

Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.